The stability of nonlinear Schr\"odinger equations with a potential in high Sobolev norms revisited

We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}


Introduction
We consider the nonlinear Schrödinger equation with a potential V iu t = −∆u + V * u + 2|u| 2 u x ∈ T d , t ∈ R, (1.1) where V is a smooth convolution potential. The system (1.1) is an infinite dimensional Hamiltonian system associated with the Hamiltonian functional H(u,ū) = T d |∇u| 2 + (V * u)ū + |u| 4 dx.
The aim of this work is to use a Hamiltonian dynamical method for studying the long time stability of small solutions in H s (T d ) for a sufficiently large s.
In this paper we consider V a random potential {σ n } being a sequence of i.i.d. random variables uniformly distributed over [− 1 2 , 1 2 ]. In usual NLS, the potential V (x) is multiplicative type, but here we choose the convolution type potential so that the formalism is simpler in Fourier variables. We assume that the potential V is an even function to ensure its Fourier coefficient v n ∈ R. We define the measure space for potential: for some m ∈ Z + . We endow W with the product probability measure, that is, As we handle small solutions, we rescale u(t, x) by (1.3) u(t, x) = εq(ε 2 t, x) = ε q n (ε 2 t)e inx .
We state our main theorem.
Theorem 1. There exists a subset V ⊂ W of full measure, such that for a given V ∈ V the following holds; for a given B > 0 there exist C, s, and ε(B) > 0 such that if u(0) H s ≤ ε 2 , the solution u to the Cauchy problem where T can be as large as ε −B .
We remark that the theorem holds true for |u| 2p u for any p ≥ 1. The analysis is similar, thus for simplicity we present p = 1 case in this paper.
Theorem 1 is a version of Birkhoff normal form theorems for the infinite dimensional Hamiltonian system: the Hamiltonian flow associated to a Hamiltonian remains close to the initial state during an arbitrarily long polynomial time (ε −B ), if the initial state has a sufficiently small by ε in H s (T d ). Some of instructive expositions for the Birkhoff normal form theory can be found in [1,6,7]. There are other stability results on (1.1) such as the KAM theorem by Elliason-Kuksin [9] with analytic potentials V , or Nekhorochev type theorem by Faou-Grebert [5] with analytic data. Theorem 1 reproves a dynamical consequence of infinite dimensional Birkhoff normal form theorem by Bambusi-Grebert [2]. In [2], the authors construct an abstract Birkhoff normal theorem to infinite dimensional Hamiltonian systems and apply it to PDEs with tame modulus. [2] is systematic and applicable to a wide range of PDE examples.
In this work, we revisit the problem with a more direct approach to the equation. In performing the Birkhoff normal form, we would like to track how the Hamiltonians are changed. Using a sequence of frequency cut-off we obtain a concrete information on the final Hamiltonian to exhibit the stability result. In fact, we are inspired by Bourgain [3], and this work follows a similar line of [3].
In [3] Bourgain consider one dimensional Schrödinger equations with random initial data, When λ = 0 (1.6) has been known to be integrable. In [8] the authors proves that the global Birkhoff coordinate exists. In [3] Bourgain proves that for given B > 0, for almost all initial data (with probability one) the solutions are stable up to time ǫ −B . The work of [3] does not rely on integrability, however the presence of cubic term |u| 2 u is essential. As like [2] and [3] uses Birkhoff normal form, a nonlinear change of coordinate of symplectic transform to reduce the non resonant terms from the given Hamiltonian. We use the formulation of the sequence of Birkhoff normal form of Bourgain's to obtain a similar result for (1.1). In use of normal forms, the nonresonancy is inherited from the randomness of initial data as opposed to in (1.1), where the nonresonancy condition is from random potentials. Indeed, the randomness is explicit in the Hamiltonian (1.5) for the (1.1) due to the random potential V . If we define ω n = |n| 2 /ε 2 + v n /ε 2 , the denominator arising in normal form transform is of the form Ω(n) = ω n 1 − ω n 2 + · · · ± ω nr n = (n 1 , n 2 , . . . , n r ).
As a similar argument to [2] (or [5]), we obtain the lower bound estimate for most of potentials V . 1 The lower bound here depends on µ(n): the third biggest entry among |n j | ′ s where n = (n 1 , n 2 , . . . , n r ), and n − denotes the least entry. For (1.6) the randomness is given to the initial data by q n (0) = ε(1 + |n|) −(1+s) σ n . Indeed, rescaling (1.6) by (1.3), one can write the associated Hamiltonian as The latter equality follows from that all the resonant terms of q n 1q n 2 q n 3q n 4 are fully resonant on T. 2 If we replace |q n | 2 by J n = |q n | 2 − |q n (0)| 2 , the randomness comes into play in the Hamiltonian: q n 1q n 2 q n 3q n 4 .
In [3] the lower bound estimate of Ω(n), holds with large probability, where n * 1 is the biggest entry of n = (n 1 , n 2 , . . . , n r ). Note that the right hand side has also the factor n −2s − . Because s is chosen to be large for the perturbation terms of Hamiltonian to be small enough, the lower bound of (1.8) becomes smaller as increasing s. This small denominater issue can be overcome if coefficients of perturbation terms are appropriately small. In [3] the author performed the normal form transformation to (1.1) inductively to have the series of Hamiltonians and to reach the final one, for which coefficients are small as desired. Once the right induction hypothesis are assumed on the size of coefficients of polynomials in Hamiltonian, the consequential analysis goes straightforward in [3].
largest frequency µ(n). For 1 dimensional (1.1) as well as (1.6) there is no difference in analysis if n * 1 and µ(n) are replaced with each other in the lower bound estimates (1.7) or (1.8). It is due to that n 1 − n 2 + · · · = 0 implies µ(n) (n * 1 ) 1/2 on T 1 . But this is no longer true for T d , d > 1. (See the estimates around (4.6) ). Another aspect of our approach is that we are able to see the regularity of the potential V with respect of B and s. Indeed, m is less than O( s B ) and s is bigger than B 3 .
We mention that the abstract theorem in [2] is applied to several other equations than (1.6) to obtain Birkhoff normal form theorems. It might be possible that the inductive use of normal form transform in [3] can be applied to reprove the known results on Birkhoff normal form theorems. We have not pushed in this direction, however the method seems quite robust. To our knowledge the similar use of iterative Birkhoff normal form transforms is found in [10]. In [10] Wang proved a long time Anderson localization for the 1-d lattice nonlinear random Schrödinger equation. We also remark that in [4] Cohen,Hairer, and Lubich proved a long time stability result for 1-d nonlinear Klein-Gordon equation via modulated Fourier expansion method without using Birkhoff normal form.
The paper is organised as follows: In Section 2, we state preliminary setting of Hamiltonian systems and Birkhoff normal form as well as the estimate of the denominator. Section 3 includes the main analysis of the Birkhoff normal form. We present the reduction of Hamiltonian and the estimate of coefficient of them. In Section 4, we provide the proof of main theorem.

Preliminaries
2.1. Symplectic transfomations. We briefly review basic definitions on the infinite dimensional Hamiltonian system. In practice what we will use in the sequel is the equations (2.3) -(2.5). For more details we refer to [6], [7].
The phase space P is defined by We identify q ∈ H s (T d ) with (q n ) ∈ l 2 s by q = q n e in·x and call (q,q) a canonical coordinate of P s . We endow P s with the symplectic 2 form which induces the symplectic operator J, Poisson bracket { , } as follows, A smooth function F ∈ C(P s , C) is called a Hamiltonian. The Hamiltonian vector field associated to F is defined by and the Hamiltonian flow associated to F by the integral curve (q(t),q(t)) along J∇F such that (q(t),q(t)) satisfies the ODE In terms of coordinate (q n ,q n ) it is written Next we introduce the symplectic transformations. A diffeomorphism ϕ : P s → P s is called symplectic transformation if ϕ preserves the Poisson bracket Symplectic transformations preserve the flow law, that is, if (q,q) is the Hamiltonian flow associated to H, the new coordinate (q ′ ,q ′ ) given by (q ′ ,q ′ ) ϕ → (q,q) satisfies the following system of ODE, What it follows we consider the symplectic transformation, time 1-shift. For a given Hamiltonian F let us consider the Hamiltonian flow generated by F, and denote the solution at time 1 by q n The Hamiltonian is shifted to H • Φ F by the coordinate change Φ F . We note that by (2.1), (2.2) and the chain rule it holds that Applying the taylor series expansion centered at t = 0, we have the following expression of Now we demonstrate how to reduce a lower order polynomials of given Hamiltonian using a time 1-shift. Back to the Hamiltonian a n q n 1q n 2 q n 3q n 4 we have a shifted Hamiltonian by a time 1-shift by F . If we choose a n q n 1q n 2 q n 3q n 4 ).
We can only reduce 'non resonant' monomials of Ω(n) = 0, meanwhile, there are abundant resonant momonials in H. This is where the randomness comes to play by modulating frequency Ω(n) so that the denominator is away from zero at a large probability.

The lower bound of the denominator.
In performing the Birkhoff normal form, we should know that the Hamiltonian has good behaved to the Poisson bracketing. As one see from (2.5), we require a lower bound of the denominators, Ω(n), to satisfy so called the strongly non resonant condition. The following lemma guarantees the strongly non resonant condition is rather generically satisfied.
Our strongly nonresonant condition is controlled by the third largest frequency and the lowest frequency, as well as the regularity parameter m of potential space. In one dimensional NLS (1.6), we have µ(n) (n * 1 ) 1/2 . Thus, the lower bound may be involved in n * 1 . However, in higher dimensional case, this is no longer true. The proof of (3.9) is similar to that in Faou and Grebert [5]. For the convenience of readers, we place it in the Appendix B.

The Poisson brackets.
We use the multi index notation as follows: A straightfoward computation shows that We denote the contraction by I ∼m m = I m /I m for m ∈ m and q ∼n n = q n /q n for n ∈ k, or q n /q n if n ∈ p. Moreover, when m = (m 1 , . . . , m i , . . . , m l ) and m i is contracted, we denote the multi index after the contraction by m ∼m i := (. . . , m i−1 , m i+1 , . . . ). So I ∼m m = I m ∼m etc. We denote the number of n appearing in m by ♯n(m) , then compute that For simplicity, we slightly abuse notations, writing In the sequel, we will estimate the coefficients c mn for each cases. Thus, the equality means that c mn of LHS is replaced by new coefficient c mn , (still denoted by c mn ), with the same upper bound.
{I m q n , I m ′ q n ′ } give rise to two types, which are occurred from loss of I m or a pair of (q n ,q n ). As above, we write (2.10)

The reduction of Hamiltonians
In this section, we discuss how to iterate the symplectic transformations and show that starting from the Hamiltonian (1.5), we reduce to the final Hamiltonian H b . Then, we show the new Hamitonian flow associated to H b , still denoted by {q n (t)}, remains ε-neighborhood of zero for a long time T with T ∼ ε −B for any given B. Define the actions of the phase variables.
Let N a and N ∞ be cut-off parameters and we set a large parameter A(> 200B). In the middle of the reduction procedure, we have the Hamiltonians of the following form: Here monomials of Σ 2 , Σ 3 , Σ 4 , and Σ 7 are of degree ≤ A. Moreover, the degree of q n in Σ 3 has at least 4. In Σ 3 ∼ Σ 7 , q n is fully nonresonant in the sense that In other words, if it were k i = p j , then the term q k iq p j = |q k i | 2 already makes I k i and is set aside from q n . The decomposition is not unique. For example, I m can be counted either in 2 or in one of 5 , 6 , 7 . We set more parameters where 5τ is a parameter that n∈Z d 1/|n| 5τ < ∞, hence 5τ > d. (3.2) is used in the proof of Proposition 3.
We use an induction argument to prove an iteration of the Birkhoff normal forms changes the initial Hamiltonian (1.5) to a final Hamiltonian H b . For this purpose, we impose induction hypotheses to coefficients of Hamiltonians and then we check that the hypotheses are still satisfied after a Birkhoff normal form reduction. We propose induction hypotheses as follows: (3.7) Note that there is no smallness hypothesis on Σ 2 1 . In fact, eventually, Σ 2 need not to be small as it is fully resonant term. However, the Poisson bracket with Σ 2 2 produces other terms Σ 3 Σ 7 . Thus, we require the hypothesis on Σ 2 2 . In Σ k , we put |n j | into the decreasing order, It follows I (m,Na) I (m ′ ,Na) = I ((m,m ′ ),Na) , Q (n,Na) Q (n ′ ,Na) = Q ((m,m ′ ),Na) . Then for the initial Hamiltonian (1.5), Now we explain on the form of (3.1) and the coefficient bounds (3.3) -(3.7). First of all, c mn is naturally bounded by product form: then the sum c mn I m q n converges due to |q n j | ≤ ε|n j | −s for each j. To obtain Theorem 1, c mn I m q n is not only to converge but also be smaller than ε −B . For this purpose, we choose large parameters A, N ∞ , such that the sum of monomials in Σ 4 , Σ 5 , Σ 6 , and Σ 7 are small. Σ 3 may contain harmful terms when the cut-off parameter N a is small. But by iteration, we push N a to larger number and to obtain the smallness from the factor N −4s 1 a . To be consistent with this, we impose the condition N a ≤ |n − |. Then N −4s 1 a Π 4 j=1 (|n j | ∧ N a ) s 1 N τ a ≥ 1 indeed, so the induction hypothesis (3.5) holds true for (1.5) for any N a . For a given parameter N a < N a+1 , we want to remove harmful nonresoant terms of N a ≤ |n − | ≤ N a+1 in Σ 3 via the Birkhoff normal form transformation. For this purpose we choose Hamiltonian for time 1-shift then by (2.7) we have Now, we explain how we proceed normal forms. We set a increasing sequence of parameters, For a fixed N a , in the middle of procedure, Hamiltonians are of the form (3.1). Then we take the Poisson bracket with F , {F, Σ k } for each k = 1, · · · , 7, and check the generated polynomials in {F, Σ k } can be put into one of Σ ′ j s by showing the corresponding induction hypothesis still holds (see (3.22)). Moreover we show H • Φ F allows the decomposition (3.1) satisfying the induction hypothesis with respect to N a (Proposition 1). In this step, Σ 3 consists of polynomials with a frequency cut-off N a+1 or that with an extra ε multiplied. We iterate the Birkhoff normal forms until all polynomials in Σ 3 with N a ≤ |n − | ≤ N a+1 are put into Σ 7 . Next, we increase the cut-off parameter N a to N a+1 and rearrange the Hamiltonian as in (3.1) with respect to N a+1 (Propositon 3). We iterate this procedure until N a reaches a sufficiently large N b , for which we will have a desired estimates on coefficients.
In the following we show how to obtain H a+1 from H a with details. It will be summarized in Proposition 3. First, we study the sums that {F, H a } generates. What it follows H stands for H a , taken off the subscript for notational simplicity.  We write the sum n∈k ♯n(k)I n − n∈p ♯n(p)I n as n∈n ♯n(n)I n , and bound them by n |n|I n . We apply the estimate of Ω(n) in (2.6) with noting that r(= degreeof q n ) ≤ A, |n − | ≤ N a+1 for the monomial in F , and obtain If n ∈ m ∩ l, then |(m, l) \ {n}| = |m| + |l| − 1. To obtain the coefficient for I m I ∼n l , we make product for m j ∈ (m, l ∼n ), and denote We estimate separately the cases of |a l | ≤ 1 and |a l | ≤ N −2s 1 a I (l,Na) . If |a l | ≤ 1, we have a Q (n,Na) ≤ εN −4s 1 a I ((m,l ∼n ),Na) Q (n,Na) (3.12) under a condition On the other hands, if |c l | ≤ N −2s 1 a I (l,Na) , we have an factor (|n| ∧ N a ) 2s 1 = N 2s 1 a due to the loss of I n , and a I ((m,l ∼n ),Na) Q (n,Na) (3.14) under a condition Note that we have an extra ε in the coefficient c (m,l ∼n )n when {F, Σ 2 } results in Σ 3 . ii-2) Σ 5 type The estimate and the condition are similar.
We postpone the case where {F, Σ 3 } generates Σ 2 in the end of the part (iv).
iv-1) Σ 3 type It is obtained from reduction of a pair of (q n ,q n ), which is the third case in (2.10). Let us estimate the coefficient bound of We have (|m j | ∧ N a ) 2s 1 N 2τ a Q (n,Na) Q 4,(n ′ ,Na) . Note that for {F, Σ 4 } to be Σ 3 , the reduced n is |n| ≥ N ∞ , obviously |n| ∧ N a = N a . And at least two n ′ j amongn ′ are n ′ j ≥ N ∞ since I m ′ q n ′ consists of Σ 4 . The right hand term is bounded by iv-2) Σ 4 type We treat the fisrt and second reducton cases in (2.10) and the third one separately. Let us estimate the coefficient bound of We have Na) . If all the three biggest index among {n, n ′ } arise in n ′ , we bound the right hand side by If some of three biggest arise in n, note that |n j | ≥ N a for n j ∈ n and newly included n ′ j is such that |n ′ j | ≥ N ∞ . Hence the extra coefficent are cancelled out in this step. We have iv-3) The case that {F, Σ 4 } generate Σ 5 term are estimated as same as Σ 3 . {F, Σ 3 } and {F, Σ 4 } can generate Σ 2 2 terms when q n = q n ′ . In the induction hypotheses (3.3)-(3.7) we see that the coefficient's bounds for Σ 3 and Σ 4 are assumed to be smaller than for Σ 2 2 ; For Σ 3 it is obvious, and for Σ 4 it is from I (m,Na) Q 4,(n,Na) ≤ N −3(s 1 +τ ) a I (m,Na) Q (n,Na) . So by estimates in (iii) and (iv), we have the coefficients of the generated Σ 2 2 term bounded by |c m | ≤ εN −2s 1 a I (m,Na) .
Overall the conditions for A, s, ε, {N a , N ∞ } is reduced to (3.2) and (3.20) So far, we have proven The point is that we have the extra ε in front of Σ 3 in Σ i when i ≥ 1 (The corresponding estimates are (3.10), (3.12), (3.14), (3.18), and (3.19)). Let us define Recalling (2.4), the Taylor series expansion formula of H • Φ F centered at t = 0, we obtain We denote {F, H} (k) := {F, · · · {F k times , H}, · · · } and {F, H} (0) = H. Under the initial condition q(0) H s ≤ ε and a consistencty condition to be proved in Section 4, the remainder converges, so we simply write

Proof.
By (3.22), note that {F, Σ 0 } cancels Σ 3 with N a ≤ |n − | < N a+1 , and we have H + {F, H} = Σ 0 + Σ 1 + Σ 3 N a+1 ≤|n − |<N∞ + 4εΣ 3 (3.24) For k ≥ 2 we assume the induction hypothesis: with 4 k k! ≤ 32 3 . The propostion follows by adding (3.24) and (3.25). So far we removed monomials of N a ≤ |n − | < N a+1 in Σ 3 and obtain an extra ε factor in front of Σ 3 . We will go on until the increasing exponent is begger than A so that we have ε A Σ 3 , which joins Σ 6 ; let us consider the normal form transform of H F by the associated Hamiltonian εF , where we use the same notation F to denote c mn Ω(n) I m q n with c mn I m q n the monomial of Σ 3 of H F . Here, Σ 3 N a+1 ≤|n − |<N a+1 means the summation of term with condition N a+1 ≤ |n − | < N a+1 . Similarly as Propostion 1 we compute H F • Φ εF as follows.
What it follows, for notational simplicity, we use Proposition 1 in the form of

Proposition 2. By the induction argument we have
Proof. First we note that Hence 4,5,6,7 .
On the other hand, we have according to the policy in front of Proposition 1. We have Note that we have Assume the induction hypothesis hold for k ≥ 2 The k = 2 case is established by same computations as above. Then it is straightforward that the k + 1-step holds: So we have as desired.
We can repeat the above procedure all over again by taking the normal form transformation with Φ ε 2 F . Denote Inductively, we have the following proposition.

Proposition 3. If k > A we have
For such k we denote H F (k) by H a+1 . The coefficients for H a+1 satisfy the induction hypothesis in (3.3) -(3.6) replacing N a by N a+1 .

Proof.
We show only the second assertion. At the time of reaching k > A the coefficients for H a+1 remain to be bounded as (3.3) -(3.6). To upgrade a to a + 1 we check (3.3) -(3.6) separately as follows.
The monomials of 2 2 satisfy |n + | ≥ N a+1 , while the other monomials goes to 2 1 for which we impose no condition. So there is at least one m j of |m j | ≥ N a+1 , and for this m j we have The monomials of Σ 3 satisfy N a+1 ≤ |n − | and the degree of q n is at least 4. We have The hypothesis (3.5) for Σ 4 is automatically upgraded. The monomials of Σ 5 are of degree bigger than A. We have Since τ = 10s A , the extra coefficient is smaller than 1. Lastly Σ 6 , Σ 7 are automatically upgraded.
We set N 1 = 1 and N 2 = N ∞ , and perform Proposition 1 to Proposition 3. Then Σ 3 is empty. (The emptiness of Σ 3 is not crucial for the following analysis). The final Hamiltonian is written as where Before proceeding to prove Theorem 1, we mension that the stability condition is preserved under the symplectic transforms. Indeed, the new Hamiltonian flow q n is obtained from a time-1 shift for the evolution iq n = ∂F ∂q n , q n (0) = q n , q n (1) = q n .
Using the definition of F we estimate that q n ∼ q n as follows: The flow {q n (t)} is given by where a mn , b mn , c mn are bounded by (3.28) -(3.31), and c mn = 0 if the degree of I m q n is less than 2A. The consistency assumption for {q n } is By (4.2), we have be treated same.

Appendix A. Summary of parameters
We introduce many parameters in the analysis. Here, for reader's convenience, we summarize the size relation of parameters. In the Theorem 1, parameter B > 0 is given. Then we consecutively choose A, s, τ, N ∞ as follows: and further m, s, A are such that m ≤ s A .
Also we choose ε = ε(C, A, s) such that In fact, the conditions imposed in Section 3 are reduced to and it can easily be verified from the above choices. Note that in this work, we need only N 1 , N ∞ .

Appendix B. Proof of Lemma 2
Lemma 2 is the direct consequence of Proposition 4. We closely follow the argument in [5]; the difference here is that the exponent of n + involves only r. In the original version in [5] it comes n + −4rm in Proposition 4.
Let us think the case in which we have two independent random variable x, y, uniformly For A not to be empty, c satisfies |c| ≤ 2M + η. Similarly, let x 1 , . . . , x n be indepenent random variables, each uniformly distributed over [−M, M ], and A = {|a 1 x 1 + a 2 x 2 + · · · + a n x n + c| ≤ η}.